On the second largest component of random hyperbolic graphs

We show that in the random hyperbolic graph model as formalized by Gugelmann et al. in the most interesting range of $\frac12 < \alpha < 1$ the size of the second largest component is $\Theta((\log n)^{1/(1-\alpha)})$, thus answering a question of Bode et al. We also show that for $\alpha=\frac12$ with constant probability the corresponding size is $\Theta(\log n)$, whereas for $\alpha=1$ it is $\Omega(n^{b})$ for some $b > 0$.